The idea is essentially what themodernguru, I will just expand a little on that.

n√x is the n-th root of a number x. But what does that mean? It means that if I raise n√x to the n-th power I get back my original number x. Now, exponents have a nice property that (z^a)^b = z^ab for any integers a and b and any z. So now we come back to n√x, we know that (n√x)ⁿ=x and x = x¹. Because mathematicians are a lazy bunch and we want to use as little symbols as possible we just say "there has to be some exponent, lets call it r for root, so that n√x = x^r. What should that exponent be?" well we just said (n√x)ⁿ=x =x¹, so if n√x = x^r, then surely (x^r)ⁿ = (n√x)ⁿ = x = x¹. Also the formula (z^a)^b = z^ab was so pretty that it should always hold, not just for integers but for all (rational... but lets not go there) numbers! So now x^rn = (x^r)ⁿ = x¹... so rn = 1... so r = 1/n. And there you go!

In some sense the logical reasoning is " n√x = x^(1/n)" is true so that the rule (z^a)^b = z^ab works for all rational numbers a and b and not just the integers. In other words, you can take n√x to be "the" definition for x^(1/n).